2006 8P FSI - jumping rpm / ignition / and transmission problem?

Hey guys, Im from Melbourne Australia, just wanted to see if anyone had the same issues with me.

First, I just bought this car (A3 8P, 2 door, FSI 2.0L) for roughly 3 weeks, I've just realised that every morning, or not driving for a few hours, when I start up the car, the rpm fluctuates around 1000-1100 for about 30-60 seconds, then it drops back to 800-900 rpm (I won't say it's a cold start coz the temperature's like 30Celcius outside??!). The car jerks a bit too during the period of time, and if I don't bother to wait and switch into R/D gear, it's really obvious that the car is shaking, but when it drops back to 900 rpm everything seems fine. It only happens in the morning/after a long time not driving...if I start it up again in an hour or so, the rpm doesn't fluctuate..

Second, it doesnt happen all the time I am not sure if its misfiring or anything, sometimes it's really normal but sometimes when I start up the car it cranks a bit longer, sometimes shorter? and I remembered once it cranked but didn't start, then the second time was alright...

Third, I went to hand wash my car and after that I realised the transmission was stuck in the third gear and not going up and down even if i slow down/speed up...switched off the engine for a few minutes, started the car up again it's alright now...not sure if this relates but now if I am pressing the pedal, say from 20km/h (2nd/3rd gear) all the way to 60km/h, rpm gets about 3000-4000 before the trasmission gets into 4th gear, or maybe into 5th gear directly from 3rd...is that normal?

I'm gonna get the car to scan and find some codes (Im expecting a few issues anyway) but hopefully it's not something major...I am just wondering if anyone can give me some insights so I can prepare on how much I need to spend